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136
On Herbrand Skeletons
, 1995
"... . Herbrand's theorem plays an important role both in proof theory and in computer science. Given a Herbrand skeleton, which is basically a number specifying the count of disjunctions of the matrix, we would like to get a computable bound on the size of terms which make the disjunction into ..."
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. Herbrand's theorem plays an important role both in proof theory and in computer science. Given a Herbrand skeleton, which is basically a number specifying the count of disjunctions of the matrix, we would like to get a computable bound on the size of terms which make the disjunction into a quasitautology. This is an important problem in logic, specifically in the complexity of proofs. In computer science, specifically in automated theorem proving, one hopes for an algorithm which avoids the guesses of existential substitution axioms involved in proving a theorem. Herbrand's theorem forms the very basis of automated theorem proving where for a given number n we would like to have an algorithm which finds the terms in the n disjunctions of matrices solely from the shape of the matrix. The main result of this paper is that both problems have negative solutions. 1 Introduction By the theorem of Herbrand we have for a quantifierfree OE: j= 9 x OE( x) iff j= OE( a 1 ) OE( ...
On the Complexity of Reasoning in Kleene Algebra
 Information and Computation
, 1997
"... We study the complexity of reasoning in Kleene algebra and *continuous Kleene algebra in the presence of extra equational assumptions E; that is, the complexity of deciding the validity of universal Horn formulas E ! s = t, where E is a finite set of equations. We obtain various levels of complexi ..."
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We study the complexity of reasoning in Kleene algebra and *continuous Kleene algebra in the presence of extra equational assumptions E; that is, the complexity of deciding the validity of universal Horn formulas E ! s = t, where E is a finite set of equations. We obtain various levels of complexity based on the form of the assumptions E. Our main results are: for * continuous Kleene algebra, ffl if E contains only commutativity assumptions pq = qp, the problem is \Pi 0 1 complete; ffl if E contains only monoid equations, the problem is \Pi 0 2 complete; ffl for arbitrary equations E, the problem is \Pi 1 1  complete. The last problem is the universal Horn theory of the *continuous Kleene algebras. This resolves an open question of Kozen (1994). 1 Introduction Kleene algebra (KA) is fundamental and ubiquitous in computer science. Since its invention by Kleene in 1956, it has arisen in various forms in program logic and semantics [17, 28], relational algebra [27, 32], aut...
Relativization of the Theory of Computational Complexity
, 1972
"... Blum's machineindependent treatment of the complexity of partial recursire functions is extended to relative algorithms (as represented by Turing machines with oracles). We prove relativizations of several results of Blum complexity theory, such as the compression theorem. A recursive relatedn ..."
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Cited by 12 (5 self)
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Blum's machineindependent treatment of the complexity of partial recursire functions is extended to relative algorithms (as represented by Turing machines with oracles). We prove relativizations of several results of Blum complexity theory, such as the compression theorem. A recursive relatedness theorem is proved, showing that any two relative complexity measures are related by a fixed recursive function. This theorem allows us to obtain proofs of results for all measures from proofs for a particular measure.
The Interactive Nature of Computing: Refuting the Strong ChurchTuring Thesis
, 2007
"... The classical view of computing positions computation as a closedbox transformation of inputs (rational numbers or finite strings) to outputs. According to the interactive view of computing, computation is an ongoing interactive process rather than a functionbased transformation of an input to a ..."
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The classical view of computing positions computation as a closedbox transformation of inputs (rational numbers or finite strings) to outputs. According to the interactive view of computing, computation is an ongoing interactive process rather than a functionbased transformation of an input to an output. Specifically, communication with the outside world happens during the computation, not before or after it. This approach radically changes our understanding of what is computation and how it is modeled. The acceptance of interaction as a new paradigm is hindered by the Strong ChurchTuring Thesis (SCT), the widespread belief that Turing Machines (TMs) capture all computation, so models of computation more expressive than TMs are impossible. In this paper, we show that SCT reinterprets the original ChurchTuring Thesis (CTT) in a way that Turing never intended; its commonly assumed equivalence to the original is a myth. We identify and analyze the historical reasons for the widespread belief in SCT. Only by accepting that it is false can we begin to adopt interaction as an alternative paradigm of computation. We present Persistent Turing Machines (PTMs), that extend TMs to capture sequential interaction. PTMs allow us to formulate the Sequential Interaction Thesis, going beyond the expressiveness of TMs and of the CTT. The paradigm shift to interaction provides an alternative understanding of the nature of computing that better reflects the services provided by today’s computing technology.
Solving Linear Equations Over Polynomial Semirings
 RUTGER UNIVERSITY (NJ
"... We consider the problem of solving linear equations over various semirings. In particular, solving of linear equations over polynomial rings with the additional restriction that the solutions must have only nonnegative coefficients is shown to be undecidable. Applications to undecidability proofs o ..."
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We consider the problem of solving linear equations over various semirings. In particular, solving of linear equations over polynomial rings with the additional restriction that the solutions must have only nonnegative coefficients is shown to be undecidable. Applications to undecidability proofs of several unification problems are illustrated, one of which, unification modulo one associativecommutative function and one endomorphism, has been a longstanding open problem. The problem of solving multiset constraints is also shown to be undecidable.
Subrecursion as Basis for a Feasible Programming Language
 Proceedings of CSL'94, number 933 in LNCS
, 1994
"... We are motivated by finding a good basis for the semantics of programming languages and investigate small classes in subrecursive hierarchies of functions. We do this with the help of pairing functions because in this way we can explore the amazing coding powers of Sexpressions of LISP within t ..."
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We are motivated by finding a good basis for the semantics of programming languages and investigate small classes in subrecursive hierarchies of functions. We do this with the help of pairing functions because in this way we can explore the amazing coding powers of Sexpressions of LISP within the domain of natural numbers. In the process of doing this we introduce a missing stage in Grzegorczykbased hierarchies which solves the longstanding open problem of what is the precise relation between the small recursive classes and those of complexity theory. 1 Introduction We investigate subrecursive hierarchies based on pairing functions and solve a longstanding open problem in small recursive classes of what is the relationship between these and computational complexity classes (see [11]). The problem is solved by discovering that there is a missing stage in Grzegorczykbased hierarchies [7, 11]. The motivation for this research comes from our search for a good programming langu...
A Survey on Embedding Programming Logics in a Theorem Prover
 Institute of Information and Computing Sciences Utrecht University
, 2002
"... Theorem provers were also called 'proof checkers' because that is what they were in the beginning. They have grown powerful, however, capable in many cases to automatically produce complicated proofs. In particular, higher order logic based theorem provers such as HOL and PVS became popula ..."
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Theorem provers were also called 'proof checkers' because that is what they were in the beginning. They have grown powerful, however, capable in many cases to automatically produce complicated proofs. In particular, higher order logic based theorem provers such as HOL and PVS became popular because the logic is well known and very expressive. They are generally considered to be potential platforms to embed a programming logic for the purpose of formal verification. In this paper we investigate a number of most commonly used methods of embedding programming logics in such theorem provers and expose problems we discover. We will also propose an alternative approach: hybrid embedding.
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
"... ..."
Communication, computability, and common interest games
 Games and Economic Behavior
, 1990
"... This paper provides a theory of equilibrium selection for oneshot twoplayer niteaction strategicform Common Interest games. A single round of costless unlimited preplay communication is allowed. Players are restricted to use strategies which are computable in the sense of Church's Thesis. ..."
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Cited by 8 (1 self)
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This paper provides a theory of equilibrium selection for oneshot twoplayer niteaction strategicform Common Interest games. A single round of costless unlimited preplay communication is allowed. Players are restricted to use strategies which are computable in the sense of Church's Thesis. The equilibrium notion used involves perturbations which are themselves computable. The only equilibrium payo vector which survives these strategic restrictions and the computable perturbations is the unique Paretoe cient one. JEL Classification: C72,C79.